Finding The Period Of The Trigonometric Function Y = (2/3)cos(4/7x) + 2

In the realm of mathematics, trigonometric functions play a pivotal role in modeling periodic phenomena. Understanding the concept of a period is crucial for analyzing and interpreting these functions. The period of a trigonometric function is the horizontal distance over which the function's graph completes one full cycle. In simpler terms, it's the length of the interval after which the function's values start repeating. This article delves into the intricacies of determining the period of a specific trigonometric function, $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$, providing a comprehensive explanation and step-by-step solution. Let's embark on this mathematical journey to unravel the secrets of periodic functions.

Understanding the General Form of Cosine Functions

Before diving into the specifics of the given function, it's essential to grasp the general form of a cosine function and how its parameters influence its period. The general form is expressed as:

$$y = A \cos(Bx - C) + D$$

Where:

• A represents the amplitude, which determines the vertical stretch of the function.
• B affects the period of the function.
• C introduces a phase shift, horizontally translating the function.
• D represents the vertical shift, moving the function up or down.

The period of the standard cosine function, $y = \cos(x)$, is $2\pi$. However, when the argument of the cosine function is multiplied by a constant, such as 'B' in the general form, the period is altered. The formula to calculate the period of a cosine function in the general form is:

$$\text{Period} = \frac{2\pi}{|B|}$$

This formula highlights the inverse relationship between the value of 'B' and the period of the function. A larger value of 'B' results in a shorter period, while a smaller value of 'B' leads to a longer period. The absolute value of 'B' is used because the period is always a positive quantity.

In the context of our given function, $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$, we can identify the corresponding parameters. The amplitude, 'A', is $\frac{2}{3}$, the coefficient 'B' is $\frac{4}{7}$, and the vertical shift, 'D', is 2. The phase shift, 'C', is 0 in this case. The amplitude and vertical shift influence the vertical characteristics of the graph, while 'B' is the key factor in determining the period, which is our primary focus here.

Identifying the Coefficient 'B' in the Given Function

The heart of finding the period lies in correctly identifying the coefficient 'B' in the given function, $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$. Comparing this function to the general form, $y = A \cos(Bx - C) + D$, we can clearly see that the term multiplying 'x' inside the cosine function is $\frac{4}{7}$. Therefore, the value of 'B' in this specific case is $\frac{4}{7}$.

It's crucial to accurately identify 'B' because it directly impacts the period calculation. A misidentification of 'B' will lead to an incorrect period value. In this function, the fraction $rac{4}{7}$ is the key to unlocking the function's period. This fraction signifies that the function's cycle is compressed compared to the standard cosine function, $y = \cos(x)$. The period will be shorter because 'B' is a fraction less than 1, effectively stretching the function horizontally. This concept is fundamental to understanding how transformations of trigonometric functions affect their periods.

Having correctly identified 'B' as $\frac{4}{7}$, we are now well-equipped to apply the period formula and determine the period of the given function. The next step involves plugging this value into the formula and performing the necessary calculations to arrive at the final answer. The accurate identification of 'B' is a critical step in the process, ensuring that the subsequent calculations are based on the correct value.

Applying the Period Formula

Now that we have identified the value of 'B' as $\frac{4}{7}$, we can proceed to apply the period formula to calculate the period of the function $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$. The period formula, as established earlier, is:

$$\text{Period} = \frac{2\pi}{|B|}$$

Substituting the value of 'B' into the formula, we get:

$$\text{Period} = \frac{2\pi}{|\frac{4}{7}|}$$

Since the absolute value of $rac{4}{7}$ is simply $rac{4}{7}$, the equation becomes:

$$\text{Period} = \frac{2\pi}{\frac{4}{7}}$$

To divide by a fraction, we multiply by its reciprocal. Therefore, the equation transforms into:

$$\text{Period} = 2\pi \times \frac{7}{4}$$

Simplifying the expression, we can cancel out the common factor of 2 between 2 and 4, resulting in:

$$\text{Period} = \pi \times \frac{7}{2}$$

Finally, we arrive at the period of the function:

$$\text{Period} = \frac{7}{2}\pi$$

This calculation demonstrates the straightforward application of the period formula once the value of 'B' is correctly identified. The period of the function $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$ is $rac{7}{2}\pi$, which signifies the length of one complete cycle of the cosine wave. Understanding this calculation is vital for analyzing the function's behavior and its graphical representation.

Interpreting the Result: Period = 7π/2

The calculated period of the function $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$ is $\frac{7}{2}\pi$. This value represents the horizontal distance required for the function's graph to complete one full cycle. In simpler terms, after every $rac{7}{2}\pi$ units along the x-axis, the pattern of the cosine wave repeats itself.

To visualize this, imagine tracing the graph of the function. Starting from any point, trace the curve until it returns to its original y-value and is about to repeat its upward or downward trajectory. The horizontal distance covered during this trace is the period, which is $\frac{7}{2}\pi$ in this case. This period is longer than the period of the standard cosine function, $y = \cos(x)$, which has a period of $2\pi$. The fractional value of 'B' ($\frac{4}{7}$) in our function caused this horizontal stretching of the graph.

The period is a fundamental characteristic of periodic functions, providing crucial information about their behavior. Knowing the period allows us to predict the function's values at various points and understand its cyclical nature. For instance, if we know the function's value at a particular 'x' value, we can predict its value at 'x' plus the period, and so on. This predictability is what makes trigonometric functions so useful in modeling real-world periodic phenomena, such as sound waves, light waves, and oscillations.

In summary, the period of $\frac{7}{2}\pi$ for the function $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$ indicates that the function's graph completes one full cycle over this interval, and this understanding is crucial for analyzing and interpreting the function's behavior.

Selecting the Correct Answer

Having meticulously calculated the period of the function $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$ as $\frac{7}{2}\pi$, we can now confidently select the correct answer from the given options. The options provided are:

A) $\frac{4}{7}\pi$ B) $\frac{7}{2}\pi$ C) $\frac{7}{4}\pi$ D) $3\pi$

Comparing our calculated period, $\frac{7}{2}\pi$, with the options, it's clear that option B, $\frac{7}{2}\pi$, matches our result. Therefore, option B is the correct answer.

This step highlights the importance of accurate calculations in mathematics. A slight error in identifying 'B' or applying the period formula would lead to an incorrect period value, and consequently, selecting the wrong answer. The methodical approach we followed, from understanding the general form of cosine functions to identifying 'B', applying the formula, and interpreting the result, ensures that we arrive at the correct solution.

In conclusion, by carefully analyzing the function and applying the period formula, we have successfully determined the period and identified the correct answer as option B, $\frac{7}{2}\pi$. This process reinforces the understanding of periodic functions and the significance of accurately calculating their periods.

Conclusion: Mastering the Concept of Period

In this comprehensive exploration, we have successfully navigated the process of finding the period of the trigonometric function $y = \frac{2}{3} \cos(\frac{4}{7}x) + 2$. We began by understanding the general form of cosine functions and the role of the coefficient 'B' in determining the period. We meticulously identified 'B' as $\frac{4}{7}$ in the given function and then applied the period formula, $ ext{Period} = \frac{2\pi}{|B|}$, to arrive at the period of $\frac{7}{2}\pi$.

Throughout this process, we emphasized the importance of accurate calculations and a thorough understanding of the underlying concepts. We also interpreted the result, explaining that the period of $\frac{7}{2}\pi$ represents the horizontal distance over which the function's graph completes one full cycle. This understanding is crucial for analyzing the behavior of trigonometric functions and their applications in modeling periodic phenomena.

The ability to determine the period of trigonometric functions is a fundamental skill in mathematics and has wide-ranging applications in various fields, including physics, engineering, and computer science. Mastering this concept allows us to analyze and predict the behavior of periodic systems, from the oscillations of a pendulum to the propagation of electromagnetic waves.

By carefully following the steps outlined in this article, readers can confidently tackle similar problems and deepen their understanding of trigonometric functions and their periods. The key takeaways include:

• Understanding the general form of trigonometric functions.
• Accurately identifying the coefficient 'B'.
• Applying the period formula correctly.
• Interpreting the result in the context of the function's graph.

With these skills, you are well-equipped to explore the fascinating world of periodic functions and their applications.